Related papers: Bogomolov conjecture over function fields for stab…

In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field.

In this paper, we prove the Effective Bogomolov's Conjecture for hyperelliptic curves defined over function fields.

The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4…

The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has established this conjecture over number fields, and Moriwaki generalized it to the…

We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose…

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…

Let M_g be the moduli space of stable curves of genus g >= 2. Let D_i be the irreducible component of the boundary of M_g such that general points of D_i correspond to stable curves with one node of type i. Let M_g^0 be the set of stable…

This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory…

We give a conjectural construction of Bridgeland stability conditions on the derived category of fibred threefolds. The construction depends on a conjectural Bogomolov-Gieseker type inequality for certain stable complexes. It can be…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author's recent theorem on equidistribution in families of abelian varieties. This generalizes…

We establish the geometric Bogomolov conjecture for semiabelian varieties over function fields. We show a closed subvariety contains Zariski dense sets of small points, if and only if, after modulo its stabilizer, it is a torsion translate…

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain…

We give an explicit uniform result on the Mordell conjecture for non-isotrivial curves over function field of characteristic 0. The proof is based on Vojta's method, and make use of Zhang's admissible adelic line bundles and a quantitative…

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such…

Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that…

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While…

We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A…

We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over Q if it holds over number fields.